The Vorticity Equation in a Rotating Stratified Fluid

Chapter 7

The Vorticity Equation in a Rotating Stratified Fluid

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The vorticity equation in one form or another and its interpretation provide a key to understanding a wide range of atmospheric and oceanic flows.

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The full Navier-Stokes' equation in a rotating frame is D

Dt u p

g

f u k u

+ ∧ = − ∇ 1 − +

ρ ν∇

where p is the total pressure and f = fk.

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We allow for a spatial variation of f for applications to flow on a beta plane.

The vorticity equation for a rotating, stratified,

viscous fluid

u⋅ ∇u =∇( u¹₂ ²)+ ∧ω u

Now

∂

∂u ρ ν∇

u f u k u

t + ∇

d i b

g

D

Dt

a

f a

f

a

f

or

Dt = − ⋅ ∇ +u f ....

Note that ∧[ ω + f] ∧u] = u⋅ (ω+ f) + (ω+ f) ⋅u - (ω+ f) ⋅ u ,

and ⋅ [ω+ f]≡0.

take the curl D

Dt u p

g

f u k u

+ ∧ = − ∇ 1 − +

ρ ν∇

ω

= ω + f is called the absolute vorticity - it is the vorticity derived in an a inertial frame

ω is called the relative vorticity, and

f is called the planetary-, or background vorticity

Recall that solid body rotation corresponds with a vorticity 2Ω.

Terminology

Dω

Dt

is the rate-of-change of the relative vorticity

Dt

a

f a

f

a

f

− ⋅ ∇ u f : If f varies spatially (i.e., with latitude) there will be a change in ω as fluid parcels are advected to regions of different f.

Note that it is really ω + f whose total rate-of-change is determined.

Interpretation

ω + ⋅ ∇ f u

a f consider first ω

u, or better still, (ω/|ω|) ⋅

.

Dt

a

f a

f

a

f

2

ρ ν∇

( ) ( )

ω⋅ ∇u= u = ω + +

n b

∂

∂

∂

∂

∂

s s u_s ∂s u_n u_b

u_nn +u_sb

unit vector along the vortex line

principal normal and binormal

directions

u + δu u

δs

the rate of relative vorticity production due to thestretchingofrelativevorticity the rate of production due to the bending(tilting, twisting, reorientation, etc.) of relativevorticity

f u u u

⋅ ∇ =f = + k z f

z f w

z

∂

∂

∂

∂

∂

h ∂ the rate of vorticity production due to the

bendingof planetaryvorticity D

Dt

a

f a

f

a

f

the rate of vorticity production due to the stretchingof planetaryvorticity

= (1/ρ)(Dρ/Dt)(ω + f) using the full continuity equation

− a ω + ∇ ⋅ f f u

Note thatthis term involves the total divergence, not just the horizontal divergence, and it is exactly zero in the Boussinesq approximation.

a relative increase in density⇒a relative increase in absolute vorticity.

sometimes denoted by B, this is the baroclinicity vector and represents baroclinic effects.

B = ∇

p

∧ ∇φ

D

Dt

a

f a

f

a

f

2

ρ ν∇

1

ρ

∇ρ∧∇p

Denote φ = ln θ = s/c

,

s = specific entropy = τ

ln p

− ln ρ + constants, where = τ

= 1 − κ.

Bis identically zerowhen the isoteric (constant density) and isobaric surfaces coincide.

ν∇

ω

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B represents an anticyclonic vorticity tendency in which the isentropic surface (constant s, φ, θ ) tends to rotate to become parallel with the isobaric surface.

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Motion can arise through horizontal variations in

temperature even though the fluid is not buoyant (in the sense that a vertical displacement results in restoring forces); e.g. frontal zones, sea breezes.

B = ∇

p

∧ ∇φ

The equations appropriate for such motions are

∂

∂

∂

∂ ρ

u u u u

f u

h

t h z

h

+ ⋅ ∇ _h+w + ∧ h = − ∇1 _hp

0= −1 + ρ

∂

∂p σ

and

Let

v u v u

, ,

z z x y

⎛ ∂ ∂ ∂ ∂ ⎞

ω = ∇⋅ u = − ⎜ ⎝ ∂ ∂ ∂ − ∂ ⎟ ⎠ Take the curl of (a)

(a)

The vorticity equation for synoptic scale

atmospheric motions

u_h⋅∇ = ∇u_h

d i

a f

∂

∂

ω ∂

∂

∂

∂ ρ

ω ω ω

ω

h f u u f

f u u

t

z z

h h h h h

h

h h

+ + ∇ ⋅ + ⋅ ∇ +

− + ⋅ ∇ + + ∇ ∧ = + ∇ρ ∧ ∇

( ) ( )

( _h ) w w 1 _hp

2

The vertical component of this equation is

h

2

u v

( f ) w ( f )

t z x y

w u w v 1 p p

y z x z x y y x

⎛ ⎞

∂ζ∂ = − ⋅∇ ζ + − ∂ζ∂ − ζ + ⎜⎝∂∂ +∂∂ ⎟⎠+

⎛∂ ∂ −∂ ∂ ⎞+ ⎛∂ρ ∂ −∂ρ ∂ ⎞

⎜∂ ∂ ∂ ∂ ⎟ ρ ∂ ∂⎜ ∂ ∂ ⎟

⎝ ⎠ ⎝ ⎠

u

where ζ = ⋅ k ω

= ∂ v / ∂ x − ∂ u / ∂ y

An alternative form is

D u v

( f ) ( f ) " "

Dt x y

⎛∂ ∂ ⎞ ⎛ ⎞ ⎛ ⎞

ζ + = − ζ + ⎜⎝∂ + ∂ ⎟⎠+⎜⎝ ⎟ ⎜⎠ ⎝+ ⎟⎠

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The rate of change of the vertical component of absolute vorticity (which we shall frequently call just the absolute vorticity) following a fluid parcel.

The term

is the divergence term

x y

⎛∂ ∂ ⎞

− ζ + ⎜⎝∂ + ∂ ⎟⎠

For a Boussinesq fluid: ∂u/∂x + ∂v/∂y + ∂w/∂z = 0 =>

u v w

( f ) ( f )

x y z

⎛ ∂ ∂ ⎞ ∂

− ζ + ⎜ ⎝ ∂ + ∂ ⎟ ⎠ = ζ + ∂

(ζ + f)∂w/∂z corresponds with a rate of production of absolute vorticity by stretching.

For an anelastic fluid (one in which density variations with height are important) the continuity equation is:

∂u/∂x + ∂v/∂y + (1/ρ

)∂(ρ

w)/∂z = 0 =>

0 0

( w)

u v 1

( f ) ( f)

x y z

⎛ ∂ ∂ ⎞ ∂ ρ

− ζ+ ⎜ ⎝ ∂ + ∂ ⎟ ⎠ = ζ+ ρ ∂

u v w

( f ) ( f )

x y z

⎛ ∂ ∂ ⎞ ∂

− ζ + ⎜ ⎝ ∂ + ∂ ⎟ ⎠ = ζ + ∂ ζ + f

w + dw w

The term

in the vorticity equation is the tilting term; this represents the rate of generation of absolute vorticity by the tilting of horizontally oriented vorticity

ω

= (−∂v/∂z, ∂u/∂z, 0) into the vertical by a non-uniform field of vertical motion (∂w/∂x, ∂w/∂y, 0) ≠ 0.

ξ ∂

= −∂v

w(x)

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The last term in the vorticity equation is the solenoidal term.

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This, together with the previous term, is generally small in synoptic scale atmospheric motions as the following scale estimates show:

∂

∂

∂

∂

∂

∂

∂

∂

ρ

∂ρ

∂

∂

∂

∂ρ

∂

∂

∂

δρ ρ

δ w

y u z

w x

v z

W H

U

L s

x p

y y

p x

p

L s

L −

NM O

QP ^{≤ =}

L −

NM O

QP ^{≤ = ×}

− −

− −

10

1 2 10

11 2

2 2 2

11 2

,

;

The sign≤ indicates that these may be overestimated due to cancellation.

End of

Chapter 7